Abstract: We consider the generalization of the Pólya urn scheme with possibly infinite many colors as introduced by Bandyopadhyay and Thacker (2016, 2017). For countable many colors, we prove almost sure convergence of the urn configuration under uniform ergodicity assumption on the associated Markov chain. The proof uses a stochastic coupling of the sequence of chosen colors with a branching Markov chain on a weighted random recursive tree as described in Bandyopadhyay and Thacker (2017). Using this coupling we estimate the covariance between any two selected colors. In particular, we reprove the limit theorem for the classical urn models with finitely many colors.